Finite mixture models are particularly useful when a population consists of distinct homogeneous subpopulations. However, these models do not satisfy the regularity conditions underlying many classical statistical methods, posing unique theoretical and computational challenges.  

This talk will first review my earlier contributions to the theory of maximum likelihood estimation and the development of a new class of hypothesis testing approaches. It will then introduce recent advances in two important areas: Gaussian mixture reduction and distributed learning for finite mixture models.  

Gaussian mixture reduction aims to approximate a high-order Gaussian mixture with a lower-order one, a problem with significant applications in statistics and machine learning. We propose a method based on composite transportation divergence, which provides an optimal reduced mixture under well-defined criteria. This approach is supported by an MM algorithm that guarantees convergence to at least a local optimum within a finite number of iterations.  

Next, we address challenges in distributed learning for finite mixture models. When statistical data is large and spread across multiple locations, initial estimates of the population distribution are typically computed locally before being aggregated at a central machine. While simple averaging is effective for many parametric models, it is inadequate for finite mixture models, where the parameter space is non-Euclidean. Here, our divergence-based aggregation method accounts for both statistical and computational complexities, leading to more reliable estimators.